Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.*;
import java.io.*;

public class Main {

  private static void solve() {
    int n = nextInt();

    double[] p = new double[n];

    for (int i = 0; i < n; i++) {
      p[i] = nextDouble();
    }

    double[][] dp = new double[n][n + 1];

    dp[0][0] = 1 - p[0];
    dp[0][1] = p[0];

    for (int i = 1; i < n; i++) {
      for (int j = 0; j <= n; j++) {
        dp[i][j] = dp[i - 1][j] * (1 - p[i]) + (j == 0 ? 0 : (dp[i - 1][j - 1] * p[i]));
      }
    }

    double ans = 0;
    for (int i = n / 2 + 1; i <= n; i++) {
      ans += dp[n - 1][i];
    }

    System.out.println(ans);
  }

  private static void run() {
    br = new BufferedReader(new InputStreamReader(System.in));
    out = new PrintWriter(System.out);

    solve();

    out.close();
  }

  private static StringTokenizer st;
  private static BufferedReader br;
  private static PrintWriter out;

  private static String next() {
    while (st == null || !st.hasMoreElements()) {
      String s;
      try {
        s = br.readLine();
      } catch (IOException e) {
        return null;
      }
      st = new StringTokenizer(s);
    }
    return st.nextToken();
  }

  private static double nextDouble() {
    return Double.parseDouble(next());
  }

  private static int nextInt() {
    return Integer.parseInt(next());
  }

  private static long nextLong() {
    return Long.parseLong(next());
  }

  public static void main(String[] args) {
    run();
  }
}